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Evaluating the Performance of the Erlang Models for Call Centers
Thomas R. Robbins
East Carolina University
College of Business
Department of Marketing and Supply Chain
3136 Bate Building
Greenville, NC 27858-4353
252-864-6687
ABSTRACT
In this paper we evaluate two queuing models used to analyze call centers; the Erlang C and Erlang A
models. The Erlang C is a simple model that ignores caller abandonment and is the most commonly
used model. The Erlang A model allows for abandonment, but performance measures are more difficult
to calculate. We compare the theoretical performance predictions of these models to a steady state
simulation model where many of the simplifying assumptions used in standard analytical models are
relaxed. Our findings support the assertion that the Erlang A model is more accurate, but we find that
in contrast to the Erlang C model, Erlang A tends to be optimistically biased. Our findings indicate that
neither model clearly dominates the other in all situations and that care must be taken to select the correct
model based on call center conditions and the intended purpose of the model.
Keywords: Call Centers, Queuing Models. Simulation Analysis
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1 Introduction
Call centers are examples of queuing systems; calls arrive, wait in a queue, and are then serviced by an
agent. Call centers are typically modeled using the M/M/N queuing model, or in industry standard
terminology - the Erlang C model. The Erlang C model makes many assumptions which are questionable
in the context of a call center environment. Specifically, the Erlang C model assumes that calls arrive at a
Poisson process with a known average rate, and that they are serviced by a defined number of statistically
identical agents with service times that follow an exponential distribution. Most significantly, Erlang C
assumes no abandonment. The model is used widely by both practitioners and academics.
Recognizing the deficiencies of the Erlang C model, many papers have advocated using alternative
queuing models and staffing heuristics which account for conditions ignored in the Erlang C model. The
most popular alternative is the Erlang A model, an extension of the Erlang C model that allows for caller
abandonment. For example, in a widely cited review of the call center literature (Gans et al., 2003) , the
authors state “For this reason, we recommend the use of Erlang A as the standard to replace the prevalent
Erlang C model.” Another widely cited paper examines empirical data collected from a call center (Brown
et al., 2005) and these authors make a similar statement; “using Erlang-A for capacity-planning purposes
could and should improve operational performance. Indeed, the model is already beyond typical current
practice (which is Erlang-C dominated), and one aim of this article is to help change this state of affairs.”
The purpose of this study is first, to examine the performance of each Erlang model under real –
world conditions, and second to evaluate the assertion that the Erlang A model is a superior model. We
conduct this analysis by performing a detailed simulation study. We develop a simulation model to predict
steady state expected system performance based on a realistic set of modeling assumptions as identified in
the literature. We compare key performance metrics from our simulation study to those predicted by the
Erlang C and Erlang A models and seek to characterize the error in the theoretical predictions. In this paper
we restrict the analysis to cases where the call center has sufficient capacity to handle all calls without
abandonment; sometimes referred to as the quality-driven and the quality and efficiency-driven (QED)
regimes (Gans et al., 2003).
Our findings confirm that the Erlang A model is indeed a more accurate model in the sense that it
makes predictions which over a wide range of input conditions result in a lower error. However, we also
find that Erlang A does not dominate Erlang C under all conditions; in other words there are situations in
which the Erlang C model provides a better estimate, even in cases where the abandonment level is non-
negligible. Furthermore, we find that while the Erlang C model tends to provide a pessimistic estimate
(i.e., the system performs better than predicted), the Erlang A model often provides an optimistic estimate.
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While it is well established that Erlang C based work force management systems tend to overstaff the call
center (Gans et al., 2003), we conclude that the use of the Erlang A model may lead to understaffing.
The remainder of this paper is organized as follows. In Section 2 we review the Erlang C and Erlang
A models and highlight the relevant literature. In Section 3 we present a general model of a steady state
call center environment and review the simulation model we developed to evaluate it. In Section 4 we
evaluate the performance of the Erlang C model, while section 5 evaluates the performance of the Erlang
A model. In Section 6 we compare the two models. We conclude in Section 7 with summary observations
and identify future research questions.
2 Queuing Models and the Associated Literature
Call centers are often modeled as queuing systems. Queuing models are used to estimate system
performance so that the appropriate staffing level can be determined in order to achieve a desired
performance metric such as the Average Speed to Answer, or the Abandonment Percentage. The most
common queuing model used for inbound call centers is the Erlang C model (Brown et al., 2005; Gans et
al., 2003). The Erlang C model (M/M/N queue) is a very simple multi-server queuing. Calls arrive
according to a Poisson process at an average rate of . By the nature of the Poisson process interarrival
times are independent and identically distributed exponential random variables with mean 1− . Calls enter
an infinite length queue and are serviced on a First Come – First Served (FCFS) basis. All calls that enter
the queue are serviced by a pool of n homogeneous (statistically identical) agents. Service times follow
an exponential distribution with a mean service time of 1 − . The steady state behavior of the Erlang C
queuing model is easily characterized. Following the derivation presented in (Gans et al., 2003) we define
the offered load (R), a unit-less quantity often referred to as the number of Erlangs, is defined as
R (1)
The offered utilization () (aka utilization, traffic intensity or occupancy) is defined as
( )N R N = (2)
The offered utilization represents the proportion of available agent time spent handling calls under the
assumption that all calls are serviced. Given the assumption that all calls are serviced, the traffic intensity
must be strictly less than one or the system becomes unstable, i.e. the queue grows without bound. This
system can be analyzed by solving a set of balance equations to find the steady state probability that all N
agents are busy, or equivalently the proportions of callers that must wait for service. The proportion of
callers that must wait prior to service is
1 1
0 0
10 1
! ! ! 1
m m mN N
m m
R R RP Wait
m m N R N
− −
= =
= − +
− (3)
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The Erlang C ProbWait function is a convex function, with values near 0 for low to moderate levels of
utilization, but increasing rapidly as utilization rises above 75%. Under Erlang C ProbWait reaches 100%
as utilization reaches 100%.
Another relevant performance measure for call centers managers is the Average Speed to Answer
(ASA).
0 | 0
1 1 10
1
ASA E Wait P Wait E Wait Wait
P WaitN
=
=
−
(4)
Like ProbWait the ASA curve is a convex function that increases very rapidly as utilization approaches
100%, but unlike ProbWait it has no upper limit.
A third important performance metric for call center managers is the Telephone Service Factor
(TSF), also called the “service level.” The TSF is the fraction of calls presented which are eventually
serviced and for which the delay is below a specified level. For example, a call center may report the TSF
as the percent of callers on hold less than 30 seconds. The TSF metric can then be expressed as
( )1
1 0 | 0
1 0N T
TSF P Wait T P Wait P Wait T Wait
P Wait e − −
= −
= − (5)
The Erlang C TSF function is concave, remaining near 100% for low to moderate levels of utilization, but
decreasing rapidly for utilization levels above 75%. TSF approaches 0 as utilization approaches 100%.
A fourth performance metric monitored by call center managers is the Abandonment Rate; the
proportion of all calls that leave the queue (hang up) prior to service. Abandonment rates cannot be
estimated directly using the Erlang C model because the model assumes no abandonment occurs.
A substantial amount of research analyzes the behavior of Erlang C model; much of it seeks to
establish simple staffing heuristics based on asymptotic frameworks applied to large call centers. Halfin &
Whitt (1981) develop a formal version of the square root staffing principle for M/M/N queues in what has
become known as the Quality and Efficiency Driven (QED) regime. Borst, Mandelbaum et al. (2004)
develop a framework for asymptotic optimization of a large call center with no abandonment. Janssen, van
Leeuwaarden et al. (2011) develops a refinement of the square root staffing heuristic by expanding the
Erlang C model. The Erlang C model makes many assumptions, several of which are not wholly accurate.
In the case of the Erlang C model several assumptions are questionable, but clearly the most problematic is
the no abandonment assumption, as even low levels of abandonment can dramatically impact system
performance (Gans et al., 2003). Many call center research papers however analyze call center
characteristics under the assumption of no abandonment (Borst et al., 2004; Gans & Zhou, 2007; Green,
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Kolesar, & Soares, 2003; Green, Kolesar, & Soares, 2001; Jennings, Mandelbaum, Massey, & Whitt, 1996;
Wallace & Whitt, 2005).
The Erlang C model assumes also that calls arrive according to a Poisson process. The interarrival
time is a random variable drawn from an exponential distribution with a known arrival rate. Several authors
assert that the assumption of a known arrival rate is problematic. Both major call center reviews have
sections devoted to arrival rate uncertainty. Brown, Gans et al. (2005) perform a detailed empirical analysis
of call center data. While they find that a time-inhomogeneous Poisson process fits their data, they also
find that arrival rate is difficult to predict and suggest that the arrival rate should be modeled as a stochastic
process. Many authors argue that call center arrivals follow a doubly stochastic process; a Poisson process
where the arrival rate is itself a random variable (Aksin et al., 2007; Chen & Henderson, 2001; Whitt,
2006c). Arrival rate uncertainty may exist for multiple reasons. Call center managers attempt to account
for these factors when they develop forecasts, yet forecasts may be subject to significant error. Robbins
(2007) compares four months of week-day forecasts to actual call volume for 11 call center projects. He
finds that the average forecast error exceeds 10% for 8 of 11 projects, and 25% for 4 of 11 projects. The
standard deviation of the daily forecast to actual ratio exceeds 10% for all 11 projects. Steckley, Henderson
et al. (2009) compare forecasted and actual volumes for nine weeks of data taken from four call centers.
They show that the forecasting errors are large and modeling arrivals as a Poisson process with the
forecasted call volume as the arrival rate can introduce significant error. Robbins, Medeiros et al. (2006)
use simulation analysis to evaluate the impact of forecast error on performance measures demonstrating the
significant impact forecast error can have on system performance. Sivan, Paul et al. (2009) develop a model
for load forecasting and evaluate it with test data from an Israeli bank.
Several papers address staffing requirements when arrival rates are uncertain or time-varying.
Bassamboo, Harrison et al. (2005) develop a model that attempts to minimize the cost of staffing plus an
imputed cost for customer abandonment for a call center with multiple customer and server types when
arrival rates are variable and uncertain. Harrison and Zeevi (2005) use a fluid approximation to solve the
sizing problem for call centers with multiple call types, multiple agent types, and uncertain arrivals. Whitt
(2006c) allows for arrival rate uncertainty as well as uncertain staffing, i.e. absenteeism, when calculating
staffing requirements. Steckley, Henderson et al. (2004) examine the type of performance measures to use
when staffing under arrival rate uncertainty. Robbins and Harrison (2010) develop a scheduling algorithm
using a stochastic programming model that is based on uncertain arrival rate forecasts. Aktekin and Ekin
(2016) develop staffing strategies for a call center model where arrival rate, service time, and abandonment
distributions are all uncertain. Heching and Squillante (2014) apply a two-stage stochastic optimization
approach to optimize service delivery centers. Roubos and Bhulai (2010) use an approximate dynamic
programing technique to control queues with time-varying parameters. Feldman, Mandelbaum et al. (2008)
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develop a staffing model to maintain time-stable performance when arrival rates are time-varying. Lima,
Maciel et al. (2014) develop a general call center evaluation approach and examine the issue of system
downtime.
The Erlang C model also assumes that the service time follows an exponential distribution. The
memoryless property of the exponential distribution greatly simplifies the calculations required to
characterize the system’s performance, and makes possible the relatively simple equations (3)-(5). If the
assumption of exponentially distributed talk time is relaxed, the resulting queuing model is the / /M G N
queue, which is analytically intractable (Gans et al., 2003) and approximations are required. However,
empirical analysis suggests that the exponential distribution is a relatively poor fit for service times. Most
detailed analysis of service time distributions find that the lognormal distribution is a better fit (Brown et
al., 2005; Gans et al., 2003; Mandelbaum A., Sakov A. , & S., 2001). Aktekin (2014) uses a Bayesian
mixture model to analyze service time for a call center when multiple caller types are present. The fit of
the Erlang C model in a call center environment is analyzed in Robbins, Medeiros et al. (2010).
Finally, the Erlang C model assumes that agents are homogeneous. More precisely, it is assumed
that the service times follow the same statistical distribution independent of the specific agent handling the
call. Empirical evidence supports the notion that some agents are more efficient than others and the
distribution of call time is dependent on the agent to whom the call is routed. In particular more experienced
agents typically handle calls faster than newly trained agents (Armony & Ward, 2008). Robbins (2007)
demonstrated a statistically significant learning curve effect in an IT help desk environment.
2.1 The Erlang A Model
Given the prevalence of caller abandonment in modern call centers, the no abandonment assumption of the
Erlang C model may be problematic. Unfortunately, models that allow for abandonment are significantly
more complex and difficult to characterize. The simplest abandonment model is the / /M M N M+ , or
Erlang A model. The model was originally presented by Palm in a 1946 paper written in Swedish. It was
presented in English in (Palm, 1957). The Erlang A model is presented in detail in Gans, Koole et al. (2003)
and (Mandelbaum and Zeltyn (2004).
Erlang A extends the Erlang C model by allowing abandonment. In the Erlang A model each caller
possesses an exponentially distributed patience time with mean 1 − . If the offered waiting time, the time a
caller with infinite patience would be required to wait, exceeds the customer’s patience time, the caller will
abandon the queue and hang up (A Mandelbaum & Zeltyn, 2004). While the exponentially distributed
patience time makes the calculations tractable, they are by no means straightforward. In particular,
calculation of the performance metrics requires an evaluation of the incomplete Gamma function
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1
0
( , ) , 0, 0
y
x tx y t e dt x y − − (6)
Details on how to calculate performance metrics for the Erlang A model are provided in Mandelbaum and
Zeltyn (2009). Following their notation, we define the basic building blocks J as
/
,
n
e nJ
=
(7)
and as
1
0
1
1
!
1
( 1)!
jn
j
n
j
n
−
=
−
=
−
(8)
We can then calculate the probability of waiting as
( )0 1J
P WaitJ
= −
+ (9)
Garnett, Mandelbaum et al. (2002) outline a method for an exact calculation of the Erlang A performance
metrics, and provide approximations based on an asymptotic analysis of the queue. Whitt (2006a) develops
deterministic fluid models to provide simple first-order performance descriptions for multi-server queues
with abandonment under heavy loads. Whitt (2016) develop diffusion approximations for the Erlang C and
Erlang A models. Knessl and van Leeuwaarden (2015) develop a transient analysis of the Erlang A model.
The inclusion of abandonment has a profound effect on the performance of the queuing system.
The impact of abandonment is discussed in detail in Garnett, Mandelbaum et al. (2002). First, the issue of
system stability is no longer a concern. In an Erlang C system the traffic intensity defined in equation (2)
must be strictly less than one for a steady state to exist; an intensity of one or more leads to an infinite queue
size. Furthermore, even very low levels of caller abandonment can dramatically alter system performance.
Abandonment has an important impact on the shape of the performance metric curves. Due to the
moderating effects of caller abandonment the curves are no longer strictly concave or convex; the ProbWait
curve for example takes on an s-shape. The probability of waiting increases at an increasing level as
congestion develops in the queue, but as congestion continues to grow abandonment levels begin to
increase. As callers exit the queue in increasing numbers the probability of waiting continues to increase,
but at a decreasing rate. The same effect applies to the ASA and TSF curves, both of which take on an s-
shape under the Erlang A model.
Comparisons of Erlang C and Erlang A models are developed in Mandelbaum and Zeltyn (2004)
and Garnett, Mandelbaum et al. (2002). Whitt (2005) examines the fit of the Erlang A model. Whitt (2006b)
examines the sensitivity of the Erlang A model to changes in the model parameters. Several papers examine
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staffing and scheduling issues in call centers where abandonment is allowed (Avramidis, Gendreau,
L'Ecuyer, & Pisacane, 2007; Bassamboo et al., 2005; Robbins & Harrison, 2010). Phung-Duc and
Kawanishi (2014) evaluate call center performance when callers abandon and later retry their call.
In order to develop a tractable model, the Erlang A model assumes an exponentially distributed
patience. Brown, Gans et al. (2005) examine abandonment and a customer’s willingness to wait in detail.
A customer’s patience is in general an unobservable metric; since only customers whose patience expires
abandon, the data is right censored. The exponential distribution of patience implies that the hazard rate
for abandonment is constant over time. Brown, Gans et al. (2005) show hazard rate graphs estimated from
empirical data for two different call types. Both graphs reveal hazard functions that are not constant; in
contrast they show a sharp peak near the origin indicating a substantial portion of customers are unwilling
to wait at all. Callers who abandon immediately are said to balk. The graphs also show another peak at 60
seconds after an announcement indicating a customer’s position in the queue. The issue of delay estimating
and announcing delay times, and the effect it can have on queues is evaluated in Ibrahim & Whitt (2008)
and (Huang, Mandelbaum, et al (2017). The hazard function shows a general decline over the range of
values plotted (0 to 400 secs.) Several other studies of patience curves have concluded that patience can be
best modeled as a Weibull distribution (Gans et al., 2003). The Weibull distribution supports a constant,
increasing, or decreasing hazard rate.
While many papers have noted the deficiencies of the Erlang C model, and advocated the use of the
Erlang A model, a systematic analysis of the error associated with each model is lacking. Our paper seeks
to close this gap in the literature.
3 Call Center Simulation
3.1 The Modified Model
In this section we present a revised model of a call center, relaxing several key assumptions discussed
previously. In our model calls arrive at a call center according to a Poisson process. Calls are forecasted
to arrive at an average rate of ̂ . The realized arrival rate is , where is a normally distributed random
variable with mean ̂ and standard deviation . The time required to process a call by an average agent
is a lognormally distributed random variable with mean 1 − and standard deviation . We define the
offered utilization ( ̂ ) as the system utilization that would occur if all calls were serviced by agents of
average productivity.
Arriving calls are routed to the agent who has been idle for the longest time if one is available. If all
agents are busy the call is placed in a FCFS queue. When placed in queue a proportion of callers will balk;
i.e. immediately hang up. Callers who join the queue have a patience time that follows a Weibull
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distribution. The Weibull distribution has two parameters, a shape parameter () and a scale parameter ().
When the shape parameter equals one the Weibull distribution is identical to the exponential distribution.
(Law, 2007). If wait time exceeds their patience time the caller will abandon. Calls are serviced by agents
who have variable relative productivity ir . An agent with a relative productivity level of 1 serves calls in
the average service time. An agent with a relative productivity level of 1.5 serves calls in 1.5 times the
average service time. Agent productivity is assumed to be a normally distributed random variable with a
mean of 1 and a standard deviation of r .
3.2 Experimental Design
In order to evaluate the performance of the Erlang C and Erlang A models against the simulation model,
we conduct a series of designed experiments. Based on the assumptions for our call center discussed
previously, we define the following set of nine experimental factors. High and low values are defined to
cover a broad range of call center scenarios. The forecasted arrival rate in the simulation is a quantity
derived from other experimental factors by
ˆ ˆ N = (10)
Given the relatively large number of experimental factors, a well-designed experimental approach is
required to efficiently evaluate the experimental region. A standard approach to designing computer
simulation experiments is to employ either a full or fractional factorial design (Law, 2007). However, the
factorial model only evaluates corner points of the experimental region and implicitly assumes that
responses are linear in the design space. Given the anticipated non-linear relationship of errors, we chose
instead to implement a Space Filling Design based on Latin Hypercube Sampling as discussed in (Santner,
Williams, & Notz, 2003). Given a set of d experimental factors and a desired sample of n points, the
experimental region is divided into nd cells. A sample of n cells is selected in such a way that the centers
of these cells are uniformly spread when projected onto each of the d axes of the design space. We chose
our design point as the center of each selected cell. This experimental design allows us to select an arbitrary
number of points for any experiment.
Factor Low High
1 Number of Agents 10 100
2 Offered Utilization ( ̂ ) 65% 95%
3 Talk Time (mins) 2 20
4 Patience Shape Factor 60 600
5 Forecast Error CV 0 .2
6 Patience Scale Factor .75 1.25
7 Talk time CV .75 1.25
8 Probability of Balking 0 .25
9 Agent Productivity Standard Deviation 0 .15
Table 1-Experimental Factors
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3.3 Simulation Model
Our call center model is evaluated using a straightforward discrete event simulation model. The purpose
of the model is to predict the long term, steady state behavior of the queuing system. The model generates
random numbers using a combined multiple recursive generator (CMRG) based on the Mrg32k3a generator
described in (L'Ecuyer, 1999). Common random numbers are used across design points to reduce output
variance. To reduce any start up bias we use a warm up period of 5,000 calls, after which all statistics are
reset. The model is then run until 25,000 calls have been serviced and summary statistics are collected.
For each design point we repeat this process for 500 replications and report the average value across
replications. Our primary analysis is based on an experiment with 1,000 design points.
The specific process for each replication is as follows. The input factors are chosen based on the
experimental design. The average arrival rate is calculated based on the specified talk time, number of
agents, and offered utilization rate according to equation (10). A random number is drawn and the realized
arrival rate is set based on the probability distribution of the forecast error. That arrival rate is then used to
generate Poisson arrivals for the replication. Agent productivities are generated using a normal distribution
with mean one and standard deviationp . Each new call generated includes an exponentially distributed
interarrival time, a lognormally distributed nominal talk time, a Weibull distributed time before
abandonment, and a Bernoulli distributed balking indicator. When the call arrives it is assigned to the
longest idle agent, or placed in the queue if all agents are busy. If sent to the queue the simulation model
checks the balking indicator. If the call has been identified as a balker it is immediately abandoned, if not
an abandonment event is scheduled based on the realized time to abandon. Once the call has been assigned
to an agent, the realized talk time is calculated as the product of the nominal talk time and the agent’s
productivity. The agent is committed for the realized talk time. When the call completes the agent
processes the next call from the queue, or if no calls are queued becomes idle. If a call is processed prior
to its time to abandon, the abandonment event is cancelled. If not, the call is abandoned and removed from
the queue. Over the course of the simulation we collect statistics on the proportion of customers forced to
wait, the average speed to answer, the abandonment rate, and the TSF defined as the proportion of total
callers serviced with a wait time less than 30 seconds. Extensive testing of our simulation model verifies
that all metrics are calculated consistently with the Erlang C and Erlang A predictions when the simulation
is configured to support those model’s assumptions.
After all replications of the design point have been executed the results are compared to the
theoretical predictions of the Erlang C and Erlang A models. In each case we calculate the error as the
difference between the theoretical value and the simulated value. For the Erlang C model we use the
standard analytical calculations using the same values of arrival rate, talk time, and the number of agents
used in the simulation. When testing against the Erlang A model the comparison is a bit more complicated.
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The first challenge is we wish to eliminate any approximation errors in our comparison, so rather than use
an approximate calculation for the Erlang A model we rerun the simulation configured to be consistent with
the Erlang A model assumptions, i.e. no balking, homogeneous agents, exponential talk time and
exponential patience. The simulation is run using common random numbers from the original simulation.
We feel that this approach allows us to focus on the error associated with the Erlang A assumptions, rather
than the numerical issues associated with estimating Erlang A performance measures. The second
challenge is how to set the patience parameter for the Erlang A calculation. Recall that this parameter is
not directly observable since data is heavily censored. Since we are attempting to fit the Erlang A model
to observed data, we approximate the Erlang A parameter as in (Gans et al., 2003) and (Brown et al.,
2005) by [ ]P Abandon E Wait = .
4 Erlang C Experimental Analysis
4.1 Summary Observations
We conducted an experiment with 1,000 design points. Based on our analysis we can make the following
summary observations:
• The Erlang C model is, on average, subject to a reasonably large error over this range of
parameter values.
• Measurement errors are strongly correlated across performance measures.
• The Erlang C model is on average pessimistically biased (the real system performs better than
predicted) but may become optimistically biased when utilization is high and arrival rates are
uncertain.
• Measurement error is high when the real system exhibits high levels of abandonment. The
error is strongly positively correlated with realized abandonment rate and predicted ASA.
• The Erlang C model is most accurate when the number of agents is large and utilization is
low.
• Errors decrease as caller patience increases.
We will now review our experimental results in more detail.
4.2 Correlation and Magnitude of Errors
The magnitude of errors generated by using the Erlang C model across our test space is high on average,
and very high in some cases. The errors across the key metrics are correlated with each other, and highly
correlated with the realized abandonment rate. Table 2 shows a correlation matrix of the errors generated
from the Erlang C model and the abandonment rate calculated from the simulation.
One of the key challenges we face when evaluating this model’s performance is choosing what
metric to evaluate. We have identified five key performance metrics for the model. The correlation analysis
shows that while there is a statistically significant correlation between the metrics, the correlation is far
from complete. Table 3 shows the summary statistics for each of the performance metrics across the design
space.
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Table 2 – Correlation Matrix for Erlang C Model
Table 3- Erlang C Error Statistics
Each statistic has a wide range of errors and those errors are imperfectly correlated. Our goal is to evaluate
the fit of the model overall, across all five metrics. To facilitate an overall assessment of model fit we
perform a cluster analysis. Specifically, we cluster all one thousand design points into 3 clusters, using a
k-means clustering algorithm. The inputs are the normalized errors in each of the five metrics. The clusters
are identified green for best, yellow for middle, and red for worst in terms of performance. Summary
metrics for the clusters are provided in Table 4. The cluster analysis shows that most of the design points,
665, are included in the low error cluster. The Erlang C model provides a good estimate for points in this
cluster, with errors generally less than 2%. The second largest group is the moderate error/yellow cluster,
with 27.8% of the observations. Here errors are becoming reasonably high, with the average error in
ProbWait forecast at nearly 18%, and an average ASA error of more than 48 seconds. The smallest group
is the high error/red which contains 6.1% of the data points. Here the Erlang C is a very poor predictor.
The average error in the ProbWait measure is nearly 35%, and the average ASA error is nearly 5 minutes.
Abandonment averages more than 8.5%, which of course is predicted to be 0 by the Erlang C model.
Table 4- Average Erlang C Errors by Cluster
Overall, the errors are generally pessimistically skewed, the system behaves in general better than
predicted. Recall that the error is calculated as the predicted value minus the observed value, so a positive
value for ProbWait means less calls waited than were predicted to wait. Similarly calls were answered faster
Simulated
Abandonment
Rate
Prob Wait
Error
ASA
Error
TSF
Error
Utilization
Error
Simulated Abandonment Rate 1.000
Prob Wait Error .867 1.000
ASA Error .766 .722 1.000
TSF Error -.880 -.987 -.759 1.000
Utilization Error .970 .861 .745 -.873 1.000
PW ECE ASA ECE TSF ECE AB ECE UT ECE
Min -8.0% -2.80 -51.6% -14.3% 0.0%
Avg 8.0% 33.40 -7.8% -2.4% 2.9%
Max 49.4% 1,098.96 3.6% 0.0% 14.0%
Skew 1.30 5.79 -1.56 -1.46 1.33
Cluster Obs PW ECE ASA ECE TSF ECE AB ECE UT ECE
Green 661 1.25% 3.30 -1.19% -1.04% 1.54%
Yellow 278 17.99% 48.28 -17.20% -4.27% 4.89%
Red 61 34.90% 291.73 -36.67% -8.59% 8.99%
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than predicted, the service level was higher than predicted, and agent utilization is lower than predicted.
Since the Erlang C model assumes no abandonment, actual abandonment is always higher than predicated.
The pessimistic nature of the Erlang C model is most clearly demonstrated by the ASA prediction. Figure
1 shows the predicted and realized ASA values across the design space with points coded based on the
cluster assignment. While the model regularly predicts calls waiting more than 30 seconds to be answered,
(26.3% of design points) a wait time in excess of 30 seconds occurs only 1 time out of 1,000.
Figure 1-Erlang C Predicted vs Actual ASA
To further illustrate the predictive performance of the Erlang C model we view the scatter plot
matrix of the five performance metric errors shown in Figure 2. From the matrix we can see that low error
(green) cluster is tightly packed, the medium error cluster is less tightly packed, and the bad (red) cluster is
spread out. This shows that when predictions are bad, they can be very bad. The 4th column of the matrix
is also illustrative. This column shows that the abandonment error, which is equivalent to the realized
abandonment rate. These plots clearly show that the error in each metric is strongly related to the
abandonment rate.
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Figure 2- Scatter Plot Matrix of Erlang C Errors
4.3 Drivers of Erlang C Error
Having established that error rates are high under the Erlang C model, we now turn our attention to
characterizing the drivers of that error. As discussed in the previous section, Erlang C errors are highly
correlated with the realized abandonment rate. The notion that abandonment is a major driver of errors in
the Erlang C model is further illustrated in Figure 3. This graph shows the error in the ProbWait measure
on the vertical axis and the abandonment rate from the simulation analysis on the horizontal axis.
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Erro
r in
Pro
bab
ility
of
Wai
t Es
tim
ate
(Th
eo
reti
cal -
Sim
ula
tion
)
Abandonment Rate from Simulation
Error in Probability of Wait Calculations vs. Abandonment
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Figure 3-Scatter Plot of Erlang C Errors and Abandonment Rate
The graph clearly shows that as abandonment increases, the error in the ProbWait measure increases as
well. The graph also reveals the optimistic errors, i.e. errors in which the system performed worse than
predicted, only occur with relatively low abandonment rates. The average abandonment rate for optimistic
predictions was .74%. The graph also reveals that significant error can be associated with even low to
moderate abandonment rates. For example, for all test points with abandonment rates of less than 5%, the
average error for ProbWait is 4.8%. For test points in which abandonment ranged between 2% and 5% the
average ProbWait error is 12.2%.
Abandonment however, is an output, not an input of our model. We now seek to determine the
relative impact of the 9 input parameters. We ran an Importance Test based on an F Test Statistic and
developed the following graph. This test examines the effectiveness for which each input can predict the
cluster for the design point.
Figure 4-Erlang C Importance Test
The utilization target is clearly the dominant factor, with the number of agents a distant second. This
makes sense given the distorting effects of abandonment. Environments with high offered utilization and
a small pool of agents are susceptible to high abandonment rates. To further analyze this phenomenon, we
ran a separate experiment, varying the number of agents from 10 to 100 for different offered utilization
rates, holding other parameters constant. The results are illustrated in Figure 5.
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Figure 5 - Erlang C ProbWait Errors by Call Center Size and Utilization
This graph demonstrates that the Erlang C model tends to provide relatively poor predictions for small call
centers. This error tends to decrease as the size of the call center increases. However, the graph also
illustrates that for busy centers the error remains high. For a very busy call center, running at 95% offered
utilization, the error rate remains at 30%, even with a pool of 100 agents. The errors tend to track with
abandonment; abandonment rates increase with utilization and decrease with the agent pool.
4.4 Conclusion
The Erlang C model is commonly applied to predict queuing system behavior in call center applications.
Our analysis shows that when we test the Erlang C model over a range of reasonable conditions, predicted
performance measures are subject to large errors. The Erlang C model works reasonably well for large call
centers with low to moderate utilization rates, but factors that tend to generate caller abandonment; i.e. high
utilization, small agent pools, and impatient callers, cause the model error to become quite large. While
the model tends to provide a pessimistic estimate, arrival rate uncertainty will either reduce that bias or lead
to an optimistic bias. It is clear that great care must be taken before using the Erlang C model to make any
calculations that require a high level of precision in a real call center environment.
5 Erlang A Experimental Analysis
5.1 Summary Observations
We utilize the same 1,000 design points for the analysis of the Erlang A model. In summary we find the
following
-10%
0%
10%
20%
30%
40%
50%
10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Erro
r in
Erl
ang
C P
rob
abili
ty o
f W
ait
Cal
cula
tio
n(T
he
ore
tica
l-Si
mu
late
d)
Number of Agents
Prob of Wait Error by Number of Agents and Offered Utilization
0.65
0.75
0.85
0.95
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• Erlang A errors are, on average relatively small.
• Errors across measures are correlated at a statistically significant level.
• Errors tend to be optimistic, i.e. the system performs worse than predicted.
• Arrival Rate uncertainty has the largest impact on Erlang A prediction error.
5.2 Correlation and Magnitude of Errors
The magnitude of errors generated by using the Erlang A model across our sample is on average relatively
low. Errors exhibit a moderately strong correlation as illustrated in Table 5. The level of correlation is
statistically significant, but less so than for the Erlang C model. ASA in particular has a much lower
correlation with ProbWait and TSF than in the Erlang C model.
Table 5 – Correlation matrix for the Erlang A Model
Correlations between measurement errors are statistically significant at the .01 level, with the magnitudes
of the correlations moderate to high. The errors in the ASA and TSF measures correlate strongly with the
realized abandonment rate, though the ProbWait error does not. Errors are correlated with the realized
abandonment, but the correlation is fairly weak. Statistics for the error in each measure are summarized in
Table 6. While the error rate can be non-trivial, they are much less than the error rates seen with the Erlang
C model. The magnitude of the error when using the Erlang A model is relatively low, but negatively
(optimistically) biased, the system behaves in general worse than predicted. Again, recall that the error is
calculated as the predicted value minus the observed value, so a negative value for ProbWait means more
calls waited than were predicted to wait. Similarly, calls were answered slightly slower than predicted, the
service level was lower than predicted, and as was the case with the Erlang C model agent utilization is
lower than predicted. The errors for ProbWait, ASA, and TSF have opposite signs as compared to the
Erlang C model and are skewed in the opposite direction.
Simulated
Abandonment
Rate
Prob Wait-
Error
ASA-
Error
TSF-
Error
Abandonment
Rate-Error
Simulated Abandonment Rate 1.000
Prob Wait-Error -.005 1.000
ASA-Error -.622 .468 1.000
TSF-Error .360 -.790 -.776 1.000
Abandonment Rate-Error -.033 .783 .432 -.823 1.000
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Table 6 - Erlang A Error Statistics
We once again perform a cluster analysis to group the data points based on a total assessment across all
errors. The clusters are once again ranked green for best, yellow for middle, and red for worst in terms of
overall performance. The results are shown in Table 7. Just under 60% of the design points are in the low
error cluster. Points in this cluster are very accurate, with the average for all percentage based metrics
within 1% and the ASA average at less than a second. The second largest group is the medium error/yellow
cluster. The errors here are much larger than the green cluster, but still relatively small, especially as
compared to the Erlang C model. The smallest group is the red or high error group. At 11.5% there are a
non-trivial number of points in this space, yet the errors remain relatively small, withProbWait and TSF
errors just under 5%. Figure 6 shows a scatter plot matrix of the 5 error measures.
Table 7 - Average Erlang A Errors by Cluster
PW EAE ASA EAE TSF EAE AB EAE UT EAE
Min -9.1% -15.05 -1.8% -2.6% -0.3%
Avg -1.3% -1.02 1.2% -0.3% 1.2%
Max 3.3% 0.83 13.2% 1.3% 4.8%
Skew -1.14 -2.87 2.05 -1.03 1.07
Cluster Obs PW ECE ASA ECE TSF ECE AB ECE UT ECE
Green 599 -0.21% -0.45 0.40% 0.01% 0.64%
Yellow 286 -2.22% -1.40 1.76% -0.59% 1.59%
Red 115 -4.53% -3.03 4.20% -1.42% 2.71%
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Figure 6-Erlang A Error Scatter Plot Matrix
Figure 7-Erlang A ProbWait Predicted vs Actual
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We examine the performance of the Erlang A prediction compared to the simulated value in Figure 7. This
scatter plot show the ProbWait value predicted by the model and that obtained from the simulation. The
scatter graph show that the error in ProbWait is generally independent of the expected wait. High and low
errors occur for low wait probabilities and high wait probabilities. The higher error conditions tend to be
optimistic errors, scenarios where the system performs worse than expected.
5.3 Drivers of Erlang A Errors
As we did with the Erlang C model, we now turn our attention to determining the drivers of the relative
error. We again run an importance test to determine which inputs are most influential in determining a
points error cluster. The results are shown in Figure 8. By far the most influential factor is Arrival Rate
Uncertainty. This issue is further illustrated in Figure 9, which shows the error in the ProbWait metric vs.
the ARCV parameter. For low values of ARCV, precise arrival rate prediction, the error is low and often
positive. As ARCV increases the error in ProbWait increases dramatically and is mostly negative. Recall
that the ARCV parameter defines the uncertainty in the arrival rate, not the overall error. Scenarios with
high ARCV correspond to conditions where the arrival rate is accurate on average, but varies from period
to period more than predicted by the Poisson distribution.
Figure 8-Erlang A Importance Test
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Figure 9- Erlang A ProbWait error vs Arrival Rate Uncertainty
5.4 Conclusion
Overall the Erlang A model is reasonably accurate model of a call center’s behavior. The average error
rates across all design points are relatively low. The model can however, under certain conditions, exhibit
significant error rates. The model is most susceptible to variability/uncertainty in the arrival rate. Unlike
the Erlang C model, the highest errors tend to be optimistic, the call center performs worse than predicted.
6 Comparing the Erlang C and A Models
6.1 Overview
In this section we compare the relative performance of the Erlang C and Erlang A models. We compare
prediction errors between the two models for each of the 1,000 points in our experimental design. Figure
10 shows a scatter plot of error in the ProbWait calculation for each observed point.
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Figure 10 – Comparing ProbWait Errors for Erlang C and A
Note the different scales; Erlang C error occurs over a range of -8% to 50%, while Erlang A error occurs
over a range of -9.1% to 3.3%. The average error of the Erlang C model is 7.96%, while the average error
from the Erlang A model is -1.28%. The overall assertion that the Erlang A model is more accurate is in
general supported by the data; the average error is smaller, and the range of errors is much smaller.
However, as the figure shows the model is not universally more accurate, the graph is configured to indicate
which model has the lower absolute error at each point. Of the 1,000 points tested, the absolute value of
the error from the Erlang A model is less than the absolute value of the Erlang C model error 63.5% of the
time, while the Erlang C model was more accurate 36.5% of the time.
One of the key observations of our analysis has been the pessimistic nature of the Erlang C estimate,
and the somewhat optimistic nature of the Erlang A estimate. To further investigate this issue, we calculate
the prediction percentile for each design point, i.e. the proportion of observations where the realized
proportion of callers waiting was less than then that predicted by each model.
Erlang C has a percentile score higher than or equal to the Erlang A score 100% of the time. The Erlang C
model is quite conservative, with a percentile value of 100 12.9% of the time. The percentile score exceeds
95 25.9% of the time. In some cases, the Erlang A has relatively low percentile scores, as low as 32.6, but
overall the Erlang A has a somewhat surprisingly high percentile value, greater than 50% in 93.5% of the
test points. This implies that even for points with an optimistic bias, performance will be better than
predicted in many cases but far worse in some cases. Due to arrival rate uncertainty, performance measures
-10%
-8%
-6%
-4%
-2%
0%
2%
4%
-20% -10% 0% 10% 20% 30% 40% 50% 60%
Erla
ng
A E
rro
r (
The
ore
tica
l -
Sim
ula
ted
)
Erlang C Error (Theoretical - Simulated)
Errors in ProbWait Predictions
Erlang A
Erlang C
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such as ProbWait are more variable than predicted by a model which assumes known arrival rates.
Furthermore, the distribution of ProbWait tends to have a relatively high positive skew.
6.2 Drivers of Relative Performance
We have seen that the Erlang A model is not universally more accurate than the Erlang C model. An
interesting question is under what conditions is the Erlang C model more accurate. To better understand
this we segregated the design points into two groups, as illustrated in Figure 10; those where the absolute
error of the Prob Wait metric I the Erlang C model was smaller, and those where the absolute error of the
Erlang A model was smaller.
Table 8 - Conditions of Model Accuracy
Table 8 summarizes this data. For each group it calculates the minimum, maximum, and average values of
each design parameter in each group. It also presents the p value generated from a simple hypothesis test
that the mean values of each group is the same. Factors such as caller patience, the shape of the patience
distribution, and the variability of talk time have little impact on which model is more accurate. Agent
Heterogeneity has a moderate impact, with higher levels of heterogeneity present in cases where Erlang A
is more accurate. Similarly, moderately longer talk times are present in the Erlang A group.
The most significant factors are the number of agents in the call center, their utilization, the uncertainty
of arrival rates, and the probably of balking. Erlang C tends to be more accurate in call centers with large
pools of agents that are moderately utilized, and arrival rates that are less certain. As we have seen, arrival
rate uncertainty tends to reduce the error in Erlang C predictions, while increasing the error in Erlang A
predictions. Higher levels of balking tend to make the Erlang A model the preferred model. Looking at
the realized abandonment rate, it is not surprising that Erlang A is more accurate when abandonment levels
are high.
Overall
Min Avg Max Min Avg Max Average p value
Num Agents 15 65.0 100 10 49.8 100 55 2.72E-19
Utilization Target 65.02% 74.74% 90.73% 65.11% 82.75% 94.99% 80% 1.47E-48
Talk Time 2.05 10.46 19.92 2.01 11.28 19.99 11 .0172
Patience 61.4 342.5 597.6 60.3 323.5 599.7 330 .0673
AR CV 0.035 0.136 0.200 0.000 0.081 0.200 0.1 5.02E-52
Talk Time CV 0.753 1.008 1.250 0.750 0.996 1.249 1 .2313
Patience Shape 0.753 1.002 1.249 0.750 0.999 1.250 1 .7370
Probability of Balking 0.01% 11.02% 24.84% 0.06% 13.28% 24.99% 12.5% 2.36E-06
Agent Heterogeneity 0.000 0.071 0.150 0.000 0.077 0.150 0.075 .0339
Abandonment 0.01% 0.92% 3.99% 0.00% 3.17% 14.29% 2.40% 3.72E-50
Erlang C Erlang A
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7 Summary and Conclusions
Erlang C is a model commonly applied to the analysis of call centers, and often used as the basis for
determining staffing level requirements. Erlang C is a relatively simple model that makes many
assumptions that are clearly suspect in the context of a call center. Many authors now advocate the use of
the more realistic and more complex Erlang A model. We have conducted a comprehensive simulation
analysis that shows that the Erlang C model is in fact subject to significant prediction error and that the
Erlang A model is on average much more accurate under steady state conditions.
However, our analysis also finds that while the Erlang C is conservative, making pessimistic
predictions most of the time, the Erlang A model often makes overly optimistic predictions. The optimistic
bias of the Erlang A model is driven in large part by arrival rate uncertainty, a condition that somewhat
paradoxically reduces the error of the Erlang C model. The results of our study suggest that care must be
taken when using the Erlang A model to make staffing decisions; particularly in cases where arrival rates
are subject to significant uncertainty and service level requirements are strict. Erlang C is the safer bet in
that staffing based on an Erlang C prediction is more likely to result in the service level being met, even if
the arrival rate is uncertain.
The analysis in this study is restricted to cases where the call center possesses the capacity to handle
all calls presented; a requirement for the Erlang C model to have a defined output. In some call center
environments staffing costs are significantly higher than the implied cost of customer delay and the number
of agents is limited so that essentially all customers must wait before receiving service and agent utilization
is close to 100%. This environment is sometimes referred to as the efficiency-driven regime (Gans et al.,
2003). In this environment the offered utilization is in excess of 100%, so the Erlang C model becomes
unstable and cannot be used to predict performance. The Erlang A model on the other hand allows for
abandonment and can be used to predict performance in this regime. A separate study is warranted to
determine the accuracy of the Erlang A model in this environment. Our analysis is also focused on long
term average performance under steady-state conditions. We do not examine the impact of shifting arrival
rates that occurs in real scenarios. In practice many call centers divide the day into a series of short intervals
and assume that steady state is achieved in each period. A further study could examine the impact of this
assumption on model accuracy.
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